4 ms·

In addition to being nicely divisible, 72 has an important advantage which most people don't realize: It's on the right side of 100 ln(2). The exact "rule" for

In addition to being nicely divisible, 72 has an important advantage which most people don't realize: It's on the right side of 100 ln(2). The exact "rule" for N% interest is N log(2) / log (1 + N/100), which has taylor series 100 log(2) + log(2)/2 N + O(N^2) ≈ 69.315 + 0.34657 N - 0.0005776 N^2 + ... For N approaching 0, the exact "rule" becomes the "rule of 100 log 2"; but for larger N increases slightly; the "rule of 72" is exactly correct for ~7.84687% interest, and for 15% interest it only gets as far as a "rule of 74.4". That said, the power series gives us a way to get a significantly more accurate result: Divide the annual percentage interest rate into 832 months, then add 4 months. For any interest rate between 1% and 40%, this result will be accurate to within 3 days.

jessriedel 10y ago
That doubles the attractiveness of this rule of thumb. Too bad the author left it out.
- kalid 10y ago
  Neat insight, I'll add it in :).
kalid 10y ago
Thanks Colin, just added :). (Aside: I love how useful text is a medium. In 5 minutes I can put in a quick insight to improve a decade-old article.)
patrec 10y ago
Maybe I'm obtuse but I'm not sure I understand what you mean by the "right side". You definitely want to use a number that's slightly larger than 100 ln(2) ~= 69.3 to account for the linear factor, but is there some inherent reason to use 72 rather than say 70 or 74, other than that assuming that 8% is some useful midpoint for the type of growth rates you're likely to be interested in?
- AnimalMuppet 10y ago
  I believe the point was, don't use 68 or 64.
- kalid 10y ago
  I think that's what he meant; round up (not down) because the next correction term in the series is positive. 72 is better (mentally) than 70 or 74 because it has more divisors.
- cperciva 10y ago
  72 is better than 70 or 74 because it's divisible by more small factors. My point was that it's fortunate that 72 is slightly more than 100 log 2 rather than slightly less; for "how long will it take for money to triple" (100 log 3 = 109.86) taking the closest number with lots of divisors would lead you to take a "rule of 108" but since 108 is slightly less than 100 log 3 instead of slightly more, it will produce larger errors for the common range of interest rates.

jessriedel 10y ago

That doubles the attractiveness of this rule of thumb. Too bad the author left it out.

kalid 10y ago
Neat insight, I'll add it in :).

kalid 10y ago

Thanks Colin, just added :). (Aside: I love how useful text is a medium. In 5 minutes I can put in a quick insight to improve a decade-old article.)

patrec 10y ago

Maybe I'm obtuse but I'm not sure I understand what you mean by the "right side". You definitely want to use a number that's slightly larger than 100 ln(2) ~= 69.3 to account for the linear factor, but is there some inherent reason to use 72 rather than say 70 or 74, other than that assuming that 8% is some useful midpoint for the type of growth rates you're likely to be interested in?

AnimalMuppet 10y ago
I believe the point was, don't use 68 or 64.
kalid 10y ago
I think that's what he meant; round up (not down) because the next correction term in the series is positive. 72 is better (mentally) than 70 or 74 because it has more divisors.
cperciva 10y ago
72 is better than 70 or 74 because it's divisible by more small factors. My point was that it's fortunate that 72 is slightly more than 100 log 2 rather than slightly less; for "how long will it take for money to triple" (100 log 3 = 109.86) taking the closest number with lots of divisors would lead you to take a "rule of 108" but since 108 is slightly less than 100 log 3 instead of slightly more, it will produce larger errors for the common range of interest rates.